There is a lot of bad information here. Let's start somewhere important:
How is the transfer line configured? There are two configurations installed by Agilent. One is the transfer line needle and strain relief nut, so that the transfer line of the HS goes through the septa into the GC inlet. The other is a direct plumbing of the HS transfer line into the inlet weldment. This is a stainless steel connection. (not recommended. ask me for details)
Secondly... suddenly stopped? Cranked up carrier pressure???
Some things do not jive here. Assuming you have the needle into septa configuration:
Check the sample probe. The is the needle point that punctures the vial. The hole for gas intake is on the side of the needle, and can get clogged easily, especially if using "Shake" set to "high". Droplets of liquid, especially high boiling solvents, will stick to the septa of the vial, and then coat the needle. Evaporation leaves solids behind and can clog the small needle hole.
Next check you transfer line needle. If "cranking the pressure" means increasing the supply pressure to the Headspace, then you may have a partial clog of the needle. Often bits of septa get stuck inside the transfer line needle. This is usually caused by puncturing the septa, with the strain relief nut tightened too much. This hardens the rubber, and the needle cores pieces away.
Your GC and headspace should have the same supply pressure. Usually about 80 psi. The transfer line flow rate, for the needle configuration, should be 50% of the column flow rate. This can be adjusted in special cases, but it is easy to cause major issues outside of this range.
Recommendations:
Use low shake. High will clog probes.
Your temperature changes are not dramatic enough to warrant a equil time. change loop equil time to zero.
Change inject time and loop fill time to 0.1 min. Increasing these times does not increase peak size. Increasing vial fill time increases risk of carryover. Increasing loop fill time, increases sample vented into the room.
Check the vial septa type. Often folks use "butyl rubber". This is not recommend for the temps you are using. Anything above 100 C should use silicon rubber.
Since you are using DMSO, the loop, transfer line, inlet, column max temp, and detector should all have a temp higher than the boiling point of DMSO.
As far as your last comments:
The GC will split correctly as long as you have glass wool in your liner to mix the sample flow with carrier, AND the transfer line flow is less than the total flow. Keep in mind there is a septa purge flow, so your transfer line should be less than the total flow minus the septa purge. Hence the 50% recommendation. In splitless mode, the peaks will split, if your transfer line flow is greater than the column flow. This is also an issue, if you have EPC for flow control on the GC, the splitless flow when the purge valve is off, travels to the EPC controller without passing through a chemical trap. (expensive repairs coming)
Since you are using DMSO, the loop, transfer line, inlet, column max temp, and detector should all have a temp higher than the boiling point of DMSO.
If you have a direct plumbed system, I'll type a another message.